Consider the join $\backslash ($ R $\backslash$bowtie S $\backslash )$ where the join predicate is $\backslash ($ R $.$ a$=$S $.$ b $\backslash ),$ given the following metadata about $\backslash ($ R $\backslash )$ and $\backslash ($ S $\backslash )$ :

$-$ Relation $\backslash ($ R $\backslash )$ contains $20,000$ tuples and has $10$ tuples per page $($block$)$

$-$ Relation $\backslash ($ S $\backslash )$ contains $5,000$ tuples and has $10$ tuples per page $($block$)$

$-$ Attribute $\backslash ($ b $\backslash )$ of relation $\backslash ($ S $\backslash )$ is the primary key for $\backslash ($ S $\backslash ),$ and every tuple in $\backslash ($ S $\backslash )$ matches $3$ tuples in $\backslash ($ R $\backslash )$

$-$ There exists an unclustered B$+-$tree index on R$.$a with height $3$

$-$ There exists a clustered B$+-$tree index on $\backslash ($ S $.$ b $\backslash )$ with height $2$

$-$ Main memory holds $5$ blocks

Answer the following questions:

$1.$ If $\backslash ($ R $\backslash$bowtie S $\backslash )$ is evaluated with a block nested loop join, which relation should be the outer relation? Justify your answer. What is the cost of the join in number of I$/$O$\text{'}$s$?$

$2.$ If $\backslash ($ R $\backslash$bowtie S $\backslash )$ is evaluated with an index nested loop join, what will be the cost of the join in number of I$/$O$\text{'}$s$?$ Which index you will use? Show your cost analysis.

$3.$ What is the cost of a plan that evaluates this query using sort$-$merge join. Show the details of your cost analysis.

$4.$ Evaluate the cost of computing $\backslash ($ R $\backslash$bowtie S $\backslash )$ using hash join assuming: i$)$ The main memory buffer can hold $202$ blocks, ii$)$ The main memory buffer can hold $4$ blocks.